Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T23:10:24.289Z Has data issue: false hasContentIssue false
  • English
  • Français

Relaxation Schemes for the M1 Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Hyperbolic equations and systems Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  15 January 2016

Teddy Pichard ,
Denise Aregba-Driollet ,
Stéphane Brull ,
Bruno Dubroca  and
Martin Frank
Teddy Pichard*
Affiliation:
Centre Lasers Intenses et Applications, Université de Bordeaux, 351 cours de la libération, Talence, 33400, France Mathematics division, Center for Computational Engineering Science, Rheinisch-Westfälische Technische Hochschule, Schinkelstrasse 2, Aachen, 52062, Germany
Denise Aregba-Driollet
Affiliation:
Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la libération, Talence, 33400, France
Stéphane Brull
Affiliation:
Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la libération, Talence, 33400, France
Bruno Dubroca
Affiliation:
Centre Lasers Intenses et Applications, Université de Bordeaux, 351 cours de la libération, Talence, 33400, France Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la libération, Talence, 33400, France
Martin Frank
Affiliation:
Mathematics division, Center for Computational Engineering Science, Rheinisch-Westfälische Technische Hochschule, Schinkelstrasse 2, Aachen, 52062, Germany
*
*Corresponding author. Email addresses:[email protected] (T. Pichard), [email protected] (D. Aregba-Driollet), [email protected] (S. Brull), [email protected] (B. Dubroca), [email protected] (M. Frank)
Get access

Abstract

Because of stability constraints, most numerical schemes applied to hyperbolic systems of equations turn out to be costly when the flux term is multiplied by some very large scalar. This problem emerges with the M1 system of equations in the field of radiotherapy when considering heterogeneous media with very disparate densities. Additionally, the flux term of the M1 system is non-linear, and in order for the model to be well-posed the numerical solution needs to fulfill conditions called realizability. In this paper, we propose a numerical method that overcomes the stability constraint and preserves the realizability property. For this purpose, we relax the M1 system to obtain a linear flux term. Then we extend the stencil of the difference quotient to obtain stability. The scheme is applied to a radiotherapy dose calculation example.

Keywords

RadiotherapyMoments modelsrelaxation modelsmethod of characteristics

MSC classification

Secondary: 82C40: Kinetic theory of gases 35L65: Conservation laws 65M25: Method of characteristics
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 1 , January 2016 , pp. 168 - 191
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aregba-Driollet, D. and Natalini, R.. Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal., 6:1973–2004, 2000.Google Scholar
[2]Aregba-Driollet, D., Natalini, R., and Tang, S.. Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. Math. Comp., 73:63–94, 2004.Google Scholar
[3]Berthon, C., Charrier, P., and Dubroca, B.. An HLLC scheme to solve the M1 model of radiative transfer in two space dimensions. Journal of Scientific Computing, 31(3):347–389,2007.Google Scholar
[4]Berthon, C., Frank, M., Sarazin, C., and Turpault, R.. Numerical methods for balance laws with space dependent flux: application to radiotherapy dose calculation. Commun. Comput. Phys., 10(5), 2011.Google Scholar
[5]Boltzmann, L.. Über die mechanische Bedeutung des zweiten Hauptsatzes der Warmetheorie. Wien. Ber., 53:195–220,1866.Google Scholar
[6]Bouchut, R. Construction of BGK models with a family of kinetic entropies for a given system of conservation law. J. Stat. Phys., 95(1-2):113–170,1999.Google Scholar
[7]Bouchut, F. Nonlinear stability of Finite Volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Birkhäuser, 2004.Google Scholar
[8]Bouchut, F, Guarguaglini, F. R., and Natalini, R.. Diffusive BGK approximations for nonlinear multidimensional parabolic equations. Indiana Univ. Math. J., 49:723–749,2000.Google Scholar
[9]Dubroca, B. and Feugeas, J.L.. Hiérarchie des modèles aux moments pour le transfert radiatif. C.R. Acad. Sci. Paris, 329:915–920,1999.Google Scholar
[10]Duclous, R., Dubroca, B., Filbet, F., and Tikhonchuk, V.T.. High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF applications. J. Comp. Phys, 228:5072–5100, August 2009.Google Scholar
[11]Duclous, R., Dubroca, B., and Frank, M.. A deterministic partial differential equation model for dose calculation in electron radiotherapy. Phys. Med. Biol., 55:3843–3857, July 2010.Google Scholar
[12]Falcone, M. and Ferretti, R.. Semi-lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods. J. Comp. Phys., 175(2):559–575, January 2002.Google Scholar
[13]Harten, A., Lax, P., and Van Leer, B.. On upstream differencing and gudonov-type schemes for hyperbolic conservation laws. SIAM Review, 25(1):35–61,1983.Google Scholar
[14]Hauck, C. and McClarren, R.. Positive PN closures. SIAM J. Sci. Comput., 32(5):2603–2626, 2010.Google Scholar
[15]Jaynes, E. T.. Information theory and statistical mechanics. Phys. Rev., 106(4):620–630,5 1957.Google Scholar
[16]Kershaw, D.. Flux limiting nature's own way. 1976.Google Scholar
[17]Larsen, E. W. and Pomraning, G. C.. The PN theory as an asymptotic limit of transport theory in planar geometry I: Analysis. Nucl. Sci. Eng., 109(1):49–75, september 1991.Google Scholar
[18]Levermore, C. D.. Relating eddington factors to flux limiters. J. Quant. Spectrosc. Radiat. Transfer, 31(2):149–160,1984.Google Scholar
[19]Levermore, C. D.. Moment closure hierarchies for kinetic theories. J. Stat. Phys., 83(5-6):1021–1065, June 1996.Google Scholar
[20]Mallet, J., Brull, S., and Dubroca, B.. An entropic scheme for an angular moment model for the classical Fokker-Planck-Landau equation of electrons. Commun. Comput. Phys., 15(2):422–450, 2014.Google Scholar
[21]Mallet, J., Brull, S., and Dubroca, B.. General moment system for plasma physics based on minimum entropy principle. submitted.Google Scholar
[22]Minerbo, G. N.. Maximum entropy eddington factors. J. Quant. Spectros. Radiat. Transfer, 20:541–545,1978.Google Scholar
[23]Natalini, Roberto. A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws. J. diff. eqns, 148(2):292–317,1998.CrossRefGoogle Scholar
[24]Pomraning, G. C.. The equations of radiation hydrodynamics. Pergamon Press, 1973.Google Scholar
[25]Russo, G. and Filbet, F.. Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics. Kinetic and Related Models, 2(1):231–250, 2009.Google Scholar
[26]Toro, E.F.. Riemann solvers and numerical methods for fluid dynamics. Springer, 1999.Google Scholar